The spectrum of geodesic balls on spherically symmetric manifolds

TL;DR

This paper investigates how the first eigenvalue of the Laplace operator on geodesic balls in spherically symmetric manifolds depends on the radius, deriving bounds, asymptotics, and explicit expansions for various cases.

Contribution

It introduces a Hadamard-type formula for the first eigenvalue's dependence on radius and provides explicit asymptotic expansions for small radii in general and specific manifolds.

Findings

Bounds for the first eigenvalue in sphere and hyperbolic space

Asymptotic sharpness of bounds as radius approaches zero

Existence of logarithmic terms in expansions for even-dimensional manifolds

Abstract

We study the Dirichlet spectrum of the Laplace operator on geodesic balls centred at a pole of spherically symmetric manifolds. We first derive a Hadamard--type formula for the dependence of the first eigenvalue $\lambda_{1}$ on the radius $r$ of the ball, which allows us to obtain lower and upper bounds for $\lambda_{1}$ in specific cases. For the sphere and hyperbolic space, these bounds are asymptotically sharp as $r$ approaches zero and we see that while in two dimensions $\lambda_{1}$ is bounded from above by the first two terms in the asymptotics for small $r$, for dimensions four and higher the reverse inequality holds. In the general case we derive the asymptotic expansion of $\lambda_{1}$ for small radius and determine the first three terms explicitly. For compact manifolds we carry out similar calculations as the radius of the geodesic ball approaches the diameter of the manifold. In the latter case we show that in even dimensions there will always exist logarithmic terms in these expansions.